Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\widehat{f}|$. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$ \| f-g\|_{L^2(\mathbb{R}^n)} \leq 2\cdot \| |\widehat{f}| - |\widehat{g}| \|_{L^2(\mathbb{R}^n)} + h_f\left( \|f-g\|^{}_{L^p(\mathbb{R}^n)}\right) + J(\widehat{f}, \widehat{g}),$$ where $1 \leq p < 2$, $h_f$ is an explicit nonlinear function depending on the smoothness of $f$ and $J$ is an explicit term capturing the invariance under translations. A noteworthy aspect is that the stability is phrased in terms of $L^p$ for $1 \leq p < 2$ while, usually, $L^p$ cannot be used to control $L^2$, the stability estimate has the flavor of an inverse Hölder inequality. It seems conceivable that the estimate is optimal up to constants.
The question of existence is treated for near-minimizers for the distance functional (or $E$-functional in the interpolation terminology) that are stable under the action of certain operators. In particular, stable near-minimizers for the couple $(L^1, L^p)$ are shown to exist when the operator is the projection on wavelets and these wavelets possess only some weak conditions of decay at infinity.
ETH Zürich（苏黎世联邦理工学院）；New York University（纽约大学）；Courant Institute for Mathematical Sciences（库朗数学科学研究所）；Instituto de Matemática Pura e Aplicada (IMPA)（巴西纯粹与应用数学研究所）
We consider the geometry associated to the ambiguities of the one-dimensional Fourier phase retrieval problem for vectors in ${\mathbb C}^{N+1}$. Our first result states that the space of signals has a finite covering (which we call the root covering) where any two signals in the covering space with the same Fourier intensity function differ by a trivial covering ambiguity. Next we use the root covering to study how the non-trivial ambiguities of a signal vary as the signal varies. This is done by describing of the incidence variety of pairs of signals with same fourier intensity function modu…
In the present paper by the Fourier transform we show that every linear differential equations of $n$-th order has a solution in $L^1(\Bbb{R})$ which is infinitely differentiable in $\Bbb{R} \setminus \{0\}$. Moreover the Hyers-Ulam stability of such equations on $L^1(\Bbb{R})$ is investigated.
Fourier Phase Retrieval for Finite Unions of Intervals
Hangzhou Normal University（杭州师范大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astronomy, radar, speech recognition, quantum mechanics and, perhaps most prominently, diffraction imaging. The mathematical study of phase retrieval problems possesses a long history with a number of beautiful and deep results drawing from different mathematical fields, such as harmonic analyis, complex analysis, or Riemannian geometry. The present paper aims to present a summary of some of these results with an emphasi…
We give an overview of some recent results on operator-valued $(L^{p},L^{q})$ Fourier multipliers and stability theory for evolution equations. The aim is to provide a relatively nontechnical introduction to the underlying ideas, emphasizing the connection between the two areas. We also indicate how operator-valued $(L^{p},L^{q})$ Fourier multipliers can be applied to functional calculus theory.
In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show …
A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phase retrieval if for all distinct vectors $x,y\in H$ the magnitude of the frame coefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y, x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). We consider the weaker condition where the magnitude of the frame coefficients distinguishes $x$ from every vector $y$ in a small neighborhood of $x$ (up to a unimodular scalar). We prove that some of the important theorems for phase retrieval hold for this local condition, where as some theorems are complete…
We consider the problem of reconstructing the missing phase information from spectrogram data $|\mathcal{G} f|,$ with $$ \mathcal{G}f(x,y)=\int_\mathbb{R} f(t) e^{-π(t-x)^2}e^{-2πi t y}dt, $$ the Gabor transform of a signal $f\in L^2(\mathbb{R})$. More specifically, we are interested in domains $Ω\subseteq \mathbb{R}^2$, which allow for stable local reconstruction, that is $$ |\mathcal{G}g| \approx |\mathcal{G}f| \quad \text{in} ~Ω\quad\Longrightarrow \quad \exists τ\in\mathbb{T}:\quad \mathcal{G}g \approx τ\mathcal{G}f \quad \text{in} ~Ω. $$ In recent work [P. Grohs and M. Rathmair. Stable Ga…
We study the phase retrieval problem for the short-time Fourier transform on the groups $\mathbb{Z}$, $\mathbb{Z}_d$ and $\mathbb{R}^d$. As is well-known, phase retrieval is possible, once the window's ambiguity function vanishes nowhere. However, there are only few results for windows which don't meet this condition. The goal of this paper is to establish new and complete characterizations for phase retrieval with more general windows and compare them to existing results. For a fixed window, our uniqueness conditions usually only depend on the signal's support and are therefore easily compreh…
We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this short note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness …
Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll f…
Phase retrieval is known to always be unstable when using a frame or continuous frame for an infinite dimensional Hilbert space. We consider a generalization of phase retrieval to the setting of subspaces of $L_2$ which coincides with using a continuous frame for phase retrieval when the subspace is the range of the analysis operator of a continuous frame. We then prove that there do exist infinite dimensional subspaces of $L_2$ where phase retrieval is stable. That is, we give a method for constructing an infinite dimensional subspace $Y\subseteq L_2$ such that there exists $C\geq 1$ so that …
